Does Japan have the constitutional and legal right to have its own army or navy?

Formally, Japan has no right: http://en.wikipedia.org/wiki/Article_9_of_the_Japanese_Constitution#Debate

Substantively, the Japanese state has taken such a right: http://en.wikipedia.org/wiki/Japan_Self-Defense_Forces#History

Politically, as the Chinese state has a great interest in Japan's capacity for aggressive war, Japan's interpretation of Article 9 both formally and substantively is closely watched by a large heavily armed state.

Security Challanges: Japan’s Defence Dilemma is an interesting document. It sheds some light on this history of balancing constitution (and anti-militaristic sentiment associated), Japan's strategic objectives and its alliance with the United States. It takes the view that Japan has been subverting Article 9 to "normalise" the state. It also points out that a major problem with extending the role of the JSDF (or "Normalising" its role) would be to "normalise" the relationship with the US and perhaps risk losing the US's large protective presence. However it also questions how much Japan's refusal to take a normal military role may already be jeopardizing its relationship with the US. It's an interesting view and seems to echo what the other answers are saying.

Here's an extract:

Normalisation by Stealth

The entire course of Japan’s post-war defence policy has been a slow march towards normalcy. In practice, Japan’s so-called ‘peace constitution’ has only delayed but not prevented progress towards normal statehood. Initially, successive Japanese governments engaged in a process of ‘revision by reinterpretation’ as a way of circumventing constitutional prohibitions. More recently ‘revision by legislation’ has been the preferred method of change. Both the Iraq deployment and the dispatch of naval forces to the Indian Ocean in support of the coalition in Afghanistan have been authorised by specific items of legislation in 2001 and 2003. These capped a series of laws enacted over the past decade, which have expanded and diversified the roles and capabilities of Japan’s Self-Defence Forces (SDF). Successive Japanese governments have accumulated a series of faits accomplis, creating the necessary precedents for the acquisition of new functions by the SDF.

_{2 Article 9 of the Japanese Constitution – the so-called ‘Peace Clause’ – has been consistently interpreted by the Japanese government as prohibiting Japan’s exercise of its right to collective self-defence (meaning military action to defend the United States or its forces) as well as Japanese participation in collective security operations under UN auspices. However, Japan’s inherent right to collective self-defence as a sovereign state has never been abrogated, and indeed is recognised under the UN charter. The prevailing interpretation of Article 9 on collective self-defence was made in 1981 by the Cabinet Legislation Bureau. It stated: ‘It is recognized under international law that a state has the right of collective self-defense, which is the right to use actual force to stop an armed attack on a foreign country with which it has close relations, even when the state itself is not under direct attack. It is therefore self-evident that since it is a sovereign state, Japan has the right of collective self-defense under international law. The Japanese government nevertheless takes the view that the exercise of the right of self-defense as authorized under Article NINE of the Constitution is confined to the minimum necessary level for the defense of the country. The government believes that the exercise of the right of collective self-defense exceeds that limit and is not, therefore, permissible under the Constitution’. Quoted in R J Samuels, ‘Constitution al Revision in Japan: The Future of Article 9’, The Brookings Institution, Center for Northeast Asian Policy, 15 December 2004, www.brookings.edu/fp/cnaps/events/20041215.pdf.}